Mini-courses

• S.Cantat (CNRS/Université de Rennes1, France)
  
  Groups of polynomial transformations
  Consider the affine space of dimension \(k>1\), say over the field of complex numbers, and the group of all its polynomial transformations with a polynomial inverse. By definition, this is the group of polynomial automorphisms of the affine space: it contains all affine transformations, but also non linear automorphisms of arbitrary large degrees. What properties of linear (finitely generated) groups remain valid in this larger group ? This will be the main topic of the mini-course.

• A.Erschler (CNRS/ENS Ulm, France)
  
  Growth and amenability of groups
  Growth function counts the number of elements of word length at most \(n\) in the word metric. While many known groups have either polynomial or exponential growth, a rich and interesting class of groups of intermediate growth is discovered in the eighties by Grigorchuk. Despite some progress in understanding such groups in recents decades, there is still some mystery about these groups and many fundamental questions remain open. Amenable groups are groups admitting an invariant finitely additive measure defined on all their subsets. Equivalent definitions can be given in terms of isoperimetric inequalities in Cayley graphs and in terms  of random walks. It is easy to see that any group of subexpenential growth is amenable. Moreover, it is known that any group of intermediate growth is amenable, but not elementary amenable. In my course I discuss known results and  open questions about growth and amenability.

• P.Haissinsky (Aix-MarseilleUniversité, France)
  
  Convergence groups
  Convergence groups form a class which appears naturally in different contexts. They are subgroups of homeomorphisms of Hausdorff compact spaces which are characterized by their dynamical properties. After going through their general properties, we will discuss some of their particular features. We may explore their relationships with hyperbolicity and quasiconformal maps, focus on groups acting on the 2-sphere and/or analyse their actions on compact subsets.

A list of references

• N.Monod (École Polytechnique Fédérale de Lausanne, Switzerland)
  
  Spaces of the third kind
  There are three model spaces: Euclidean, spherical and hyperbolic. In many branches of analysis and geometry, we use the first kind without specifying any dimension, and indeed often in infinite dimensions: this is called Hilbert spaces. Likewise, for the second kind, Hilbert spheres are very familiar since they underly unitary representations. In this mini-course, we study the third kind: Hyperbolic spaces of arbitrary (typically infinite) dimension. This is in particular interesting from the viewpoint of group actions.
  

Junior speakers

• Federica Fanoni (CNRS/Université de Strasbourg)
  
  Big mapping class groups and their actions on graphs
  Mapping class groups of finite-type surfaces (surfaces with finitely generated fundamental group) form an interesting class of finitely generated groups. If we consider infinite-type surfaces instead (surfaces whose fundamental group is not finitely generated), the associated mapping class groups are not finitely generated and thus not adapted to being investigated via classical geometric group theory techniques. I will discuss how one could still study these groups and show that in many cases they still act nicely on interesting graphs associated to the surface. Based on joint works with M. Durham and N. Vlamis and with T. Ghaswala and A. McLeay.
  
  
• Mikołaj Frączyk(IAS Princeton, USA)
  
  Growth of mod-p homology groups in higher rank lattices
  Let Gamma be either a higher rank lattice or finite index subgroup of the mapping class group of higher genus surface. I will describe two approaches for estimating the dimensions of mod-p homology groups in terms of the covolume or the index of Gamma in the ambient group. First approach is homotopical in nature and involves building “small” CW-complexes computing low homology groups while the second in purely geometric and requires the understanding of minimal representatives of homology classes. Talk will be party based on a joint work (in progress) with M. Abert, N. Bergeron and D. Gaboriau. 
  
  
• Radhika Gupta (University of Bristol, UK)
  
  Non-uniquely ergodic arational trees in the boundary of Outer space
  The mapping class group of a surface is associated to its Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arational trees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called "non-uniquely ergodic". I will first talk about laminations on surfaces and then present a construction of non-uniquely ergodic phenomenon for arational trees.
  This is joint work with Mladen Bestvina and Jing Tao.
  
  
• Yair Hartman (Ben Gurion University, Israel)
  
  Which groups have bounded harmonic functions?
  Bounded harmonic functions on groups are closely related to random walks on groups. It has long been known that all virtually nilpotent groups are "Choquet-Deny groups": these groups cannot support non-trivial bounded harmonic functions. Equivalently, their Furstenberg-Poisson boundary is trivial, for any random walk. I will present a recent result where we complete the classification of discrete countable Choquet-Deny groups, proving a conjecture of Kaimanovich-Vershik. We show that any finitely generated group which is not virtually nilpotent, is not Choquet-Deny. Surprisingly, the key here is not the growth rate, but rather the algebraic infinite conjugacy class property (ICC). This is joint work with Joshua Frisch, Omer Tamuz and Pooya Vahidi Ferdowsi.
  
  
• Adrien Le Boudec (CNRS/ENS Lyon, France)
  
  Commensurated subgroups and micro-supported actions
  A subgroup \(\Lambda\) of a group \(\Gamma\) is commensurated if all the conjugates of \(\Lambda\) are commensurate.  After providing a basic introduction to this notion, we will state a theorem that relates the commensurated subgroups of a finitely generated group \(\Gamma\) with the topological dynamics of the minimal and micro-supported actions of \(\Gamma\) on compact spaces. As an application we obtain a criterion to exclude the existence of non-trivial commensurated subgroups in certain classes of groups. Examples include topological full groups of amenable groups, or branch groups acting on rooted trees.
  
  Although the theorem is about finitely generated discrete groups, we will try to highlight the role played in the proof by non-discrete locally compact groups. Time permitting, we might also discuss how the notion of uniformly recurrent subgroups (URS) comes into play. This is joint work with Pierre-Emmanuel Caprace.
  
  
• Beatrice Pozzetti (Rheinische Friedrich-Wilhelms-Universität Bonn, Germany)
  
  Surface subgroups of semisimple Lie groups
  I will discuss various geometric properties of discrete subgroups of semisimple Lie groups \(G\), isomorphic to the fundamental group of a hyperbolic surface. After discussing the differences between the case \(G={\rm SL}(2,\mathbb R)\), where we will recover Teichmüller theory, and \(G={\rm SL}(2,\mathbb C)\) where we will encounter fundamental groups of quasi-Fuchsian manifolds and their limits, I will explain how for \({\rm SL}(n,\mathbb R)\), and more generally in higher rank, new interesting intermediate phenomena arise. Based on joint work with Sambarino and Wienhard and with Beyrer.
  
  
  

Research statements